00:01
In this problem it is given that g is the group of symmetries of a circle.
00:08
We are asked to prove that g has every elements of finite order as well as it has elements of infinite order.
00:18
Now if g is the symmetric group of circles, then g will have elements corresponding to the rotation of the circle through any angle theta since the rotation of a circle through any angle will be a symmetry of the circle.
00:41
Now let r denotes an element that is obtained by rotating the circle through an angle of 360 degree divided by n where n is any positive integer, that is n greater than 0 and n belongs to the set of integers z.
01:01
Then we know that rotation through 360 degree retains the original position of the circle.
01:09
Therefore, the rotation through 360 degree is the identity element of the group g.
01:17
Now, consider a group g with an identity element e.
01:24
For any element a of this group, if there exists a positive integer n such that a to the power n, is equal to e, then the smallest such positive integer is said to be the order of the element in the group.
01:43
And if no such positive integer exists, then the order of the element is said to be infinite...